Uniformly Rotating Homogeneous and Polytropic Rings in Newtonian Gravity

Abstract

An analytical method is presented for treating the problem of a uniformly rotating, self-gravitating ring without a central body in Newtonian gravity. The method is based on an expansion about the thin ring limit, where the cross-section of the ring tends to a circle. The iterative scheme developed here is applied to homogeneous rings up to the 20th order and to polytropes with the index n=1 up to the third order. For other polytropic indices no analytic solutions are obtainable, but one can apply the method numerically. However, it is possible to derive a simple formula relating mass to the integrated pressure to leading order without specifying the equation of state. Our results are compared with those generated by highly accurate numerical methods to test their accuracy.

18 pages, 12 figures. This paper is essentially a combined version of arXiv:0802.0078 and arXiv:0802.0081 and was published in this form. (v2: minor corrections)

Cited by in corpus (7)

• Orbital perturbations due to massive rings
• Toroidal figures of equilibrium from a 2nd-order accurate, accelerated SCF-method with subgrid approach
• The equilibrium of over-pressurised polytropes
• Pseudo-Newtonian Models for the Equilibrium Structures of Rotating Relativistic Stars
• Uniformly Rotating Homogeneous Rings in post-Newtonian Gravity
• The exterior gravitational potential of toroids
• A Roche Model for Uniformly Rotating Rings